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Slopes and gradients

In mathematics the slope of a straight line is the ratio of the change in y to the change in x

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For a curved line the slope at any point on the line is the slope of the tangent to the line at that point.

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In each case the slope is the tangent of the angle the straight line makes with the horizontal.

But for thousands of years people we now call civil engineers, people who build aquaducts and canals and roads and railways etc, have always used gradients rather than slopes.

Today when we think of aquaducts we usually think of the huge stone arched structures the Romans built to carry water across valleys, many of which are still standing today, in many parts of Europe and North Africa. These carried not drinking water but bath water, that is, water for the great Roman Bathhouses. The aquaduct was in fact the whole channel in which the water ran, from the spring in the mountain to the town, it is just that two thousand years later only the massive stone arches used to carry the aquaduct across a valley have survived.

These arches were needed because an aquaduct must be downhill every centimetre of the way: if it is not downhill for even a centimetre the water just overflows over the sides. This means that every part of the route must be very carefully planned.

Two thousand years before the Romans the people of Mesopotamia (the then very fertile land between the River Tigris and the River Euphrates, approximately modern Iraq) were building aquaducts (canals) to carry water from the mountains and hills to irrigate their fields.

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The problem was, they could not measure the green distances because they were either underground or up in the air. But they could measure the blue distance quite easily, with a trundle wheel, and the red distance by having a man stand at the bottom of the hill with a long pole held vertically and a man at the top of the hill with a sighting tube and a spirit level - surveyors still use this method four thousand years later. The gradient of a hill is the ratio of the height gained to the distance along the slope, that is, the sine of the angle it makes with the horizontal. Originally gradients were usually expressed as a unit fraction, for example one in fifteen, but today they are more usually given as a percentage - you can see them on road signs warning you about a steep hill. A cliff has a gradient of 100% - positive if you are climbing up it and negative if you are falling off it!
© Barry Gray May 2018